Can Rewriting P -> Q as ~Q -> ~P Lead to a False Conclusion?

Date: 01/21/2006 at 00:43:08
From: Eddie
Subject: P -> Q <> ~Q -> ~P
A teacher told the month of her birthday (let's call it M) to Peter,
and the day of her birthday (let's call it D) to John. Then, the
teacher showed a set of dates, and asked Peter and John which one was
her birthday:
4 Mar, 5 Mar, 8 Mar
4 Jun, 7 Jun
1 Sep, 5 Sep
1 Dec, 2 Dec, 8 Dec
Peter said, "If I don't know the answer, John doesn't know, too."
John said, "At first I really didn't know, but now I know."
Peter said, "I know, too."
What is the teacher's birthday?
The first time I thought the answer was 1 Sep. Peter said John
couldn't know the teacher's birthday so Peter must have Mar or Sep,
since all the days in those two months appear more than once in the list.
Of course, John can also figure out that Peter has either Mar or
Sep by Peter's argument. John knows the day and since he now knows
the answer it can't be 5 since that day happens in both Mar and Sep.
That means the day must be 1, 4 or 8. Lastly, Peter said he now also
knows the answer. That would happen if and only if it was Sep since
Mar would still have two possibilities, 4 and 8. So her birthday must
be 1 Sep.
However, I know 'if P then Q' is equivalent to 'if Not Q then Not P'.
But when I make that change, I get another date for the answer.
"If I don't know the answer, John doesn't know, too" is changed into
"If John knows the answer, I know the answer."
Since only 7 and 2 directly let John know the answer, Peter could only
have either Jun or Dec to make the argument true. But 7 or 2 can't be
the day since John says he didn't know at first. Now that John knows
it's either Jun or Dec, though, he knows the answer, so he must have
1, 4, or 8. But Peter also knows, so that can only happen if it's Jun
since Dec still has two choices. So her birthday must be 4 Jun.
I don't know why there will be two answers when I just change the
first argument by a valid conversion: P -> Q = ~Q -> ~P. Or am I
wrong somewhere in the process?
Thank you for your help.

Date: 01/21/2006 at 05:35:58
From: Doctor Jacques
Subject: Re: P -> Q <> ~Q -> ~P
Hi Eddie,
Disturbing, isn't it? At first sight, your two arguments are both
correct.
The point is that we cannot blindly use propositional logic here. In
propositional logic, a proposition is a statement that is either true
or false, and we assume that, if the same proposition is used more
than once, all occurrences have the same meaning (the same truth
value).
However, in this case, a statement like "X knows the answer" does not
satisfy that definition: such a statement can be false at some time
and true some time later (this is the whole point of the riddle: at
the beginning, nobody knows the answer, and, at the end, both Peter
and John know it).
If we were to analyze the argument in a strictly formal way, we should
consider several propositions like:
XKn : X (J, P) knows the answer at step n (1,2,3)
XSn : X says that he knows the answer at step n
and a few more, like "X knows at step m that Y does not know at
step n"...
We would also need to write down some implied statements explicitly,
for example:
XK1 -> XK2 (once you know the answer, you don't forget it)
XSn -> XKn (our guys don't lie)
in addition to the statements that can be derived from the set of
possible dates and the statements of Peter and John.
I didn't try to write that down in full (this could be interesting):
the problem can be solved in a more informal way.
I believe your first answer is the correct one. Let us be a little
more explicit:
At step 1, we know that Peter does not know the answer, since each
possible month has more than one day, and nothing else has been said.
By Peter's first statement (and the rule of modus ponens), this
implies that John does not know the answer either, and the day cannot
be 2 or 7.
We also learn that Peter knows that John does not know the answer;
this means that Peter knows that the day is neither 2 nor 7, and this
rules out the months in which these days appear, i.e., Jun and Dec.
At step 2, John first confirms that he did not know the answer (this
does not tell us anything new). Now, John has just learned that the
month is either Mar or Sep, and this is sufficient for him to deduce
the answer. This means that the day (known to John) must appear
exactly once in Mar or Sep, which leaves 1, 4, and 8.
At step 3, Peter has learned that the day is 1, 4, or 8, and this is
enough for him to know the answer. This means that exactly one of
these days can appear in the month (known to Peter), and this rules
out Mar, leaving only 1 Sep as the answer.
Now, what is wrong with the second argument?
It is quite correct that Peter could equivalently have said at step 1:
"If John knows the answer, I know the answer"
but this simply means that either John does not know the answer
(meaning the day is neither 2 nor 7, which is true), or Peter knows
the answer (which we know is false).
However, the way you interpret it in your argument is as if Peter had
said:
"If I knew John knows the answer, I would know it too" (*)
and this is completely different: the antecedent of that statement is
not "John knows the answer", but "Peter knows that John knows the
answer".
By the way, it is interesting to note that, after step 2, Peter does
indeed know that John knows the answer, and statement (*) would be
true if Peter had said it at that time. The point is that statement
(*) implies that the day must appear only once in the remaining
possible months, and this has a different meaning at step 1 and step
2.
Does this help? Please feel free to write back if you want to discuss
this further.
- Doctor Jacques, The Math Forum
http://mathforum.org/dr.math/

Date: 01/21/2006 at 10:14:52
From: Eddie
Subject: Thank you (P -> Q <> ~Q -> ~P )
Thanks, Doctor Jacques.
Now I understood how my interpretation was wrong. Thank you for your
help. But I do have two questions. First, you say:
"We also learn that Peter knows that John does not know the answer;
this means that Peter knows that the day is neither 2 nor 7, and
this rules out the months in which these days appear, i.e., Jun and
Dec."
After I discussed this with some friends, we don't understand why Jun
and Dec can be ruled out in Step 1. Isn't it possible that Peter got
Jun and said the same 1st statement of "If I don't know the answer,
John doesn't know, too."?
Second, is a "deductive" or "inductive" argument being assumed in this
question and would that cause a different answer? Should it be
restricted to deductive?
Thank you for your help.

Date: 01/23/2006 at 03:11:00
From: Doctor Jacques
Subject: Re: P -> Q <> ~Q -> ~P
Hi Eddie,
First, we know--and everybody knows--that Peter does not know the
answer at the beginning: the only thing Peter knows is the month,
and, whichever month it is, there is always a choice of more than one
day.
Assuming Peter does not lie, we have the argument:
"If P. does not know, J. does not know"
"P. does not know"
--------------------------------------
therefore,
"J. does not know"
Note that there is no problem of ambiguity here: "X does not know the
answer" means the same thing throughout the argument.
Now, this argument tells us two things:
(1) the conclusion itself--John does not know. This would be true
if anybody else (for example the teacher) has made that
statement.
(2) the fact that the statement is made by Peter.
(1) tells us that the day (known to John) cannot be 2 or 7, since in
either of these cases, there would only be one possible answer, and
John would know it.
Note that, as far as John is concerned, we have a "necessary and
sufficient condition": John knows the answer in step 1 if and only if
the day is 2 or 7.
(2) tells us that, not only John does not know the answer, but Peter
can be sure of it. Because of (1), this means that Peter knows that
the day is neither 2 nor 7, and he can only be certain of that if
neither of these days appears in the month he knows.
For example, if the month is Jun, the day can only be 4 or 7. It is
true that, if the day were 4, John would not know the answer, and
Peter's first statement would still be true. However, from Peter's
point of view, it is also possible that the day is 7, in which case
John would know the answer, and Peter's statement would be false. This
means that, if the month is Jun, it is possible that John does not
know, but Peter cannot be certain of it, and cannot therefore make
his statement.
In fact, all this shows that there is an additional hidden assumption
in this kind of problem.
We already assumed that the players do not lie, meaning their
statements are true propositions (assuming we make them explicit
enough to specify who knows what at what time).
In addition, we must assume that they are certain of what they are
saying; formally, this would mean that their statements are not only
true (which could happen by chance) but are also provable from the
information available to the speaker at the time they are made; to
put it otherwise, the players only assert things that they know to be
true, not merely things that are true in fact. This is the assumption
we use in the second part of the argument. In some similar problems
(there are many of them...), this is made explicit by a statement
like "the players are truthful and infinitely clever".
Concerning your question about inductive and deductive reasoning,
this is a different matter. In this case, we use only deductive
reasoning (even to a pathological extent...).
Inductive reasoning is the process by which you "guess" a general
rule from given known facts (observations). This is mainly how
physics (and many other disciplines, like engineering, medicine, ...)
work: we make measurements, use them to guess a plausible law (this
is inductive reasoning), derive consequences of these laws (this is
deductive reasoning), and make experiments to check those
consequences against the real world.
In mathematics also, you can use inductive reasoning (this is not
proof) in some cases. In some cases, you can guess a general formula
(by inductive reasoning) from a few values computed numerically. Once
you know what you should obtain, this may give you a clue about how
to prove it (by deductive reasoning).
Beware that "inductive reasoning" is not the same as "mathematical
induction".
- Doctor Jacques, The Math Forum
http://mathforum.org/dr.math/

Date: 01/24/2006 at 20:50:56
From: Eddie
Subject: Thank you (P -> Q <> ~Q -> ~P )
Dear Doctor Jacques,
You're so nice. We've really learned some new things, and one of us
started to get interested in mathematics again.
Thank you very much.